Past exam of the physics A course of the University of Cambridge / 2018 / Paper 2 / Section B / B9 / e

Place the join at $x=0$, with an incident wave travelling from medium 1 towards medium 2. WriteContinuity of displacement and slope gives $A+B=C$ and $k_1(A-B)=k_{2}C$. Hence reflection at a density step on a string givesFor $\boxed{{\rho }_2>{\rho }_1}$, $r<0$: the reflected transverse displacement has a phase reversal of $\pi$. For ${\rho }_2<{\rho }_1$, $r>0$ and there is no displacement-phase reversal. The transmitted displacement has no phase reversal for positive real mechanical impedances.

The reflected and transmitted mean-power fractions areFor strictly positive finite linear mass densities, $\mathcal{T}>0$; a finite density step cannot produce exact total reflection. The ideal limits areThe first behaves as a fixed end; the second as a free end. In the free-end limit the transmitted displacement tends to $2A$, but its mechanical impedance tends to zero, so it carries vanishing power. Thus total reflected energy does not require a vanishing boundary displacement. “Entirely reflected” at a merely large density contrast is an approximation.

There is no reflected wave when the characteristic mechanical impedances match:for the same tension. Then $r=0$, $C/A=1$ and $\mathcal{T}=1$. If tensions were independently adjustable, the general condition would be $T_1{\rho }_1=T_2{\rho }_2$, together with transverse-force matching; equality of densities alone would no longer be the universal criterion.